In the world's busiest airports, how is separation managed between departures and arrivals when there are only two parallel runways like in London Heathrow. What is the minimum safe separation?
With two runways, usually, one is dedicated to arrivals, and the other is dedicated to departures. That's the simplest way to handle separation -- only in-trail separation is involved.
That's usually the way traffic is managed at Heathrow (LHR) and Dubai (DXB), two of the world's busiest two-parallel-runway airports. At Heathrow, when in a westward flow (70% of the time) the runway functions are swapped about 3pm, to equalize noise impacts.
San Francisco (SFO) has two parallels, 28L and 28R, that are 750 ft apart -- very close. Under visual conditions, both runways are used for VISUAL approaches. Generally, aircraft arrive side-by-side, for wake avoidance.
For about an hour a day. 6am to 7am, when arrivals dominate, LHR allows arrivals on both runways. I don't know what separation rules are used. At SFO, under marginal conditions, both 28s are used for arrivals but aircraft are staggered and capacity is less than for visual operations.